The number of polynomials of the form `x^3 + ax^2 + bx + c` which are divisible by `x^2 + 1` and where a, b , c belong to {1,2,....,10}

The number of factors of the polynomial (x^(2)+bx+c) is

Express the following eqautions in the form of ax^(2)+bx+c=0, where a, b, c are real numbers and ane0 .

Determine the value of k for which the polynomial (2x^(4)-3x^(3)-x^(2)+x+k) is divisible by (2x - 1).

If the equations x^2 + 2x + 3 = 0 and ax^2 + bx + + c = 0 have common root, then a : b : c is _____

Determine the values of a and b when the polynomial (ax^(3)+bx^(2)+x-6) is divided by (x - 2), the remainder is 4 and (x + 2) is a factor of the polynomial.

If (x^(2)-2) be a factor of the polynomial (ax^(3)-bx^(2)-4) , find the values of a and b.

Number of points where f(x)=| x sgn(1-x^2)| is non-differentiable is a. 0 b. 1 c. 2 d. 3

If ax^(2)+ bx +c and bx ^(2) + ax + c have a common factor x +1 then show that c=0 and a =b.

Euler's substitution: Integrals of the form intR(x, sqrt(ax^(2)+bx+c))dx are claculated with the aid of one of the following three Euler substitutions: i. sqrt(ax^(2)+bx+c)=t+-x sqrt(a)if a gt 0 ii. sqrt(ax^(2)+bx+c)=tx+-x sqrt(c)if c gt 0 iii. sqrt(ax^(2)+bx+c)=(x-a)t if ax^(2)+bx+c=a(x-a)(x-b) i.e., if alpha is real root of ax^(2)+bx+c=0 (xdx)/(sqrt(7x-10-x^(2))^3) can be evaluated by substituting for x as